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Author: Heida, Martin × Subject: Random conductance model ×

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    Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Flegel, Franziska; Heida, Martin; Slowik, Martin
    We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian’s Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence.
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    On quenched homogenization of long-range random conductance models on stationary ergodic point processes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Heida, Martin
    We study the homogenization limit on bounded domains for the long-range random conductance model on stationary ergodic point processes on the integer grid. We assume that the conductance between neares neighbors in the point process are always positive and satisfy certain weight conditions. For our proof we use long-range two-scale convergence as well as methods from numerical analysis of finite volume methods.